Cédric Arhancet is a French mathematician working in the preparatory cycle for engineering schools at Lycée Lapérouse (France). He works in several areas of functional analysis including noncommutative Lp-spaces, Fourier multipliers, semigroups of operators and noncommutative geometry. More recently, he has connected his research to Quantum Information Theory. Christoph Kriegler is a German-French mathematician working at Universit Clermont Auvergne, France. His research interests lie in harmonic and functional analysis. In particular, he works on functional calculus for sectorial operators, and spectral multipliers in connection with geometry of Banach spaces on the one hand, and on the other hand on noncommutative Lp espaces and operator spaces.

• - 1. Introduction. - 2. Preliminaries. - 3. Riesz Transforms Associated to Semigroups of Markov Multipliers. - 4. Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators. - 5. Locally Compact Quantum Metric Spaces and Spectral Triples. - A. Appendix: Lévy Measures and 1-Cohomology.

Cédric Arhancet is a French mathematician working in the preparatory cycle for engineering schools at Lycée Lapérouse (France). He works in several areas of functional analysis including noncommutative Lp-spaces, Fourier multipliers, semigroups of operators and noncommutative geometry. More recently, he has connected his research to Quantum Information Theory. Christoph Kriegler is a German-French mathematician working at Universit Clermont Auvergne, France. His research interests lie in harmonic and functional analysis. In particular, he works on functional calculus for sectorial operators, and spectral multipliers in connection with geometry of Banach spaces on the one hand, and on the other hand on noncommutative Lp espaces and operator spaces.

• - 1. Introduction. - 2. Preliminaries. - 3. Riesz Transforms Associated to Semigroups of Markov Multipliers. - 4. Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators. - 5. Locally Compact Quantum Metric Spaces and Spectral Triples. - A. Appendix: Lévy Measures and 1-Cohomology.

Cédric Arhancet is a French mathematician working in the preparatory cycle for engineering schools at Lycée Lapérouse (France). He works in several areas of functional analysis including noncommutative Lp-spaces, Fourier multipliers, semigroups of operators and noncommutative geometry. More recently, he has connected his research to Quantum Information Theory. Christoph Kriegler is a German-French mathematician working at Universit Clermont Auvergne, France. His research interests lie in harmonic and functional analysis. In particular, he works on functional calculus for sectorial operators, and spectral multipliers in connection with geometry of Banach spaces on the one hand, and on the other hand on noncommutative Lp espaces and operator spaces.

• - 1. Introduction. - 2. Preliminaries. - 3. Riesz Transforms Associated to Semigroups of Markov Multipliers. - 4. Boundedness of H∞ Functional Calculus of Hodge-Dirac Operators. - 5. Locally Compact Quantum Metric Spaces and Spectral Triples. - A. Appendix: Lévy Measures and 1-Cohomology.